Over the last several decades, many mesh generation methods and a plethora of adaptive methods for solving differential equations have been developed. In this talk, we take a general approach for describing the mesh generation problem, which can be considered as being in some sense equivalent to determining a coordinate transformation between physical space and a computational space. Some new theoretical results are given that provide insight into precisely what is accomplished using mesh equidistribution (which is a standard adaptivity tool used in practice). As well, we discuss two general types of moving mesh methods for solving time dependent PDEs, those based upon a variational formulation of the mesh generation problem and those which target mesh velocity. Among the methods in the latter class are those which solve the Monge-Ampere equation and the optimal mass transport problem, an area which has seen intense research activity of late.